0 An,n1 Where ai,j is any element in Fp . Because any combinations of elements from Fp will make A invertible, we have a bection from H to (Fp...

0 An,n1 Where ai,j is any element in Fp . Because any combinations of elements from Fp will make A invertible, we have a bection from H to (Fp )n(n1)/2 . So |H| = pn(n1)/2 which makes it a Sylow p-subgroup of GLn (Fp ). Exercise (4.5.40). Prove that the number of Sylow p-subgroups of GLn (Fp ) is p + 1. Proof.